Waves such as electromagnetic, seismic and acoustic waves propagate through media in a predictable way in accordance with well understood physical principles. However, directly determining wave propagation through a medium having complex boundaries in accordance with physical laws is computationally expensive.
In one example, it may be desirable to perform non-destructive testing (NDT) in the form of ultrasonic testing on a workpiece to ensure that no defects such as inclusions are present in the raw material prior to carrying out further machining. FIG. 1 shows a cross sectional view through part of a forging 10 for a turbine disc of a gas turbine engine. The turbine disc is annular, having a shape shown in FIG. 2. The forging 10 includes an initial workpiece profile 12 (also known as a “Condition of Supply”, or COS) which is machined from a larger casting geometry 11. The workpiece 12 comprises a polygon having rectilinear sides. The workpiece 12 is further machined to form a final profile 14, which is the required shape of the turbine disc. Before the workpiece 12 is further machined, the workpiece 12 is placed within an ultrasonic inspection apparatus 15 shown in FIG. 2. The apparatus 15 contains sound conducting media such as water 16 which has a different refractive index than the workpiece 12 due to the lower speed of sound in water. An ultrasonic transducer 18 is used to perform ultrasonic testing on the workpiece 12 to determine whether parts of the final profile 14 which will be subject to high stresses include any defects.
In order to inspect the workpiece 12 to the satisfaction of regulatory authorities, it must be shown that the final profile 14 can be subjected to three separate ultrasonic testing passes per side of the workpiece profile 12. When designing the workpiece profile 12 therefore, NDT requirements require that the final profile 14 can be tested from angles defined by the external profile of the workpiece 12.
Methods are known for simulating wave propagation in order to, for instance, model ultrasonic testing of an article. On a physical level, wave propagation is governed by a wave equation, which is a partial differential equation. Methods for solving this equation to simulate wave propagation across a domain of interest over time include methods such as the Finite Difference Method (FDM), the Finite Element Method (FEM). FDM and FEM are known as “meshed” methods, and operate by discretising the problem domain into a plurality of sub-domains (or cells) known as a mesh. In the case of FDM, the differential operators in the wave equations are replaced with the difference approximations from the Taylor series. A further wave based method comprises the Boundary Element Method (BEM).
“Meshless” methods are also known, such as the Distributed Point Source Method (DPSM). In this method, the transducer face and interfaces are modelled as distributed point sources. The field variables of the ultrasonic wave are computed using Green's function. Based on Green's function, an arbitrary point in space will be influenced by the distributed points surrounded by it. Mesh free methods are preferable in some situations to meshed methods, as they have high computational efficiency. One type of Meshless method is known as a “Domain type” method. In Domain type methods, the differential equations are approximated at the boundary as well as in the domain. Strong or weak formulations of the wave equation are used in the domain area, as this type of solution is well behaved in the domain. However, in known methods, singularities (i.e. computational errors) occur at boundaries as well as at defect areas, making it difficult to distinguish boundaries from defects in the model.
Another method of solving partial differential equations such as wave equations is known as “Particle-in-Cell” (PIC). In this method, individual particles representative of field variables of the wave front are tracked in continuous phase space in a Langrangian frame, whereas the equations of motion are integrated in a computational grid known as a cell. The particles do not interact (i.e. collide) with each other. The main disadvantage of PIC is that it is computationally expensive due to the large background mesh. Increasing the size of the cells will result in faster computation, but will also result in a solution having lower accuracy. This is due to the bi-directional computation between the cell and particle layers, wherein the cell properties are calculated in part from the results of the particle layer from the previous time step, and vice versa.
In each case there is therefore a compromise between wave propagation simulation accuracy and computational expense, as more accuracy generally leads to longer computational time required to carry out the calculation. Other methods include geometrical methods such as ray tracing (described in Alpkocak and Sis, 2010), the Image Source Method (ISM, described in (Allen and Berkley, 1979) and beam tracing (described in Drumm and Lam, 2000). However, while faster, such geometrical methods are less accurate.
Consequently, there is a desire to provide a new method of determining wave propagation through media such as metals in order to, for example, determine the available scan coverage of an article. Such a method must be computationally efficient, as well as highly accurate. The present invention seeks to provide such a method.